MASx52: Assignment 3
1. Consider the binomial model with r =
1
11 ,
d = 0.9, u = 1.2, s = 100 and time steps t = 0, 1, 2.
(a) Draw a recombining tree of the stock price process, for time t = 0, 1, 2.
(b) Find the value, at time t = 0, of a European call option that gives its holder the option
to purchase one unit of stock at time t = 2 for a strike price K = 90. Write down the
hedging strategy that replicates the value of this contract, at all nodes of your tree.
You may annotate your tree from (a) to answer (b).
P
2. Let Sn = ni=1 Xi , be the simple symmetric random walk, in which (Xi )i∈N is a sequence of
i.i.d. random variables with common distribution P[Xi = 1] = P[Xi = −1] = 12 .
Let Fn = σ(S1 , . . . , Sn ), and note that (Fn )n∈N is a filtration. Which of the following
stochastic processes are previsible? Which are adapted? (Justification is not required.)
(a) An = 1{Sn ≥ 0}
(b) Bn = E[Sn | Fn−1 ]
(c) Cn = min{S1 , S2 , . . . , Sn−1 }
(d) Dn = max{S1 , S2 , . . . , Sn+1 }
3. Let Z be a random variable taking values in [1, ∞) and for n ∈ N define
Xn =
(
Z
if Z ∈ [n, n + 1)
0
otherwise.
(a) Show that, if Z ∈ L1 , then E[Xn ] → 0 as n → ∞.
(b) Let Z be the continuous random variable with probability density function
f (x) =
(
x−2
0
if x ≥ 1,
otherwise.
Show that Z is not in L1 , but that E[Xn ] → 0.
(c) Comment on what (a) and (b) tell us about the dominated converge theorem.
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